
16Stewart Shapiro. Secondorder languages and mathematical practice. The journal of symbolic logic, vol. 50 , pp. 714–742 (review)Journal of Symbolic Logic 54 (1): 291293. 1989.

16The Statue within: An Autobiography. François Jacob, F. Philip (review)Philosophy of Science 58 (1): 132132. 1991.

40Robert L. Martin and Peter W. Woodruff. On representing ‘trueinL' in L. Philosophia , vol. 5 no. 3 , pp. 213–217.  Saul Kripke. Outline of a theory of truth. The journal of philosophy, vol. 72 , pp. 690–716.  Anil Gupta. Truth and paradox. Journal of philosophical logic, vol. 11 , pp. 1–60.  Hans G. Herzberger. Notes on naive semantics. Journal of philosophical logic, vol. 11 , pp. 61–102 (review)Journal of Symbolic Logic 50 (4): 10681071. 1985.

27Stewart Shapiro. Philosophy of mathematics. Structure and ontology. Oxford University Press, New York and Oxford 1997, x + 279 pp (review)Journal of Symbolic Logic 64 (2): 923926. 1999.

1Mathematical StructuralismCambridge University Press. 2018.The present work is a systematic study of five frameworks or perspectives articulating mathematical structuralism, whose core idea is that mathematics is concerned primarily with interrelations in abstraction from the nature of objects. The first two, settheoretic and categorytheoretic, arose within mathematics itself. After exposing a number of problems, the book considers three further perspectives formulated by logicians and philosophers of mathematics: sui generis, treating structures as a…Read more

67Carnap* RepliesThe Monist 101 (4): 388393. 2018.In an imagined dialogue between two figures called “Carnap*” and “Quine*” that appeared in the Library of Living Philosophers volume in 1986, certain proposals and clarifications of the linguistic doctrine were offered by Carnap* answering Quinean objections, but these were brushed aside rather breezily in a reply to this dialogue in the same volume by Quine himself. After a brief summary of the questions at issue in that earlier dialogue, Carnap* is here allowed a final reply, introducing yet a…Read more

Varieties of Continua: From Regions to Points and BackOxford University Press. 2018.Hellman and Shapiro explore the development of the idea of the continuous, from the Aristotelian view that a true continuum cannot be composed of points to the now standard, entirely punctiform frameworks for analysis and geometry. They then investigate the underlying metaphysical issues concerning the nature of space or spacetime.

26Hilary Putnam’s Contributions to Mathematics, Logic, and the Philosophy ThereofThe Harvard Review of Philosophy 24 117119. 2017.

Mathematics without Numbers. Towards a ModalStructural InterpretationTijdschrift Voor Filosofie 53 (4): 726727. 1991.

106Mathematics Without Numbers: Towards a ModalStructural InterpretationOxford University Press. 1989.Develops a structuralist understanding of mathematics, as an alternative to set or typetheoretic foundations, that respects classical mathematical truth while ...

119Physicalism: Ontology, determination and reductionJournal of Philosophy 72 (October): 55164. 1975.

127Structuralism without structuresPhilosophia Mathematica 4 (2): 100123. 1996.Recent technical developments in the logic of nominalism make it possible to improve and extend significantly the approach to mathematics developed in Mathematics without Numbers. After reviewing the intuitive ideas behind structuralism in general, the modalstructuralist approach as potentially classfree is contrasted broadly with other leading approaches. The machinery of nominalistic ordered pairing (BurgessHazenLewis) and plural quantification (Boolos) can then be utilized to extend the c…Read more

71On the significance of the BuraliForti paradoxAnalysis 71 (4): 631637. 2011.After briefly reviewing the standard settheoretic resolutions of the BuraliForti paradox, we examine how the paradox arises in set theory formalized with plural quantifiers. A significant choice emerges between the desirable unrestricted availability of ordinals to represent wellorderings and the sensibility of attempting to refer to ‘absolutely all ordinals’ or ‘absolutely all wellorderings’. This choice is obscured by standard set theories, which rely on type distinctions which are obliter…Read more

100What is categorical structuralism?In Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.), The Age of Alternative Logics, Springer. pp. 151161. 2006.

98Mathematical Pluralism: The Case of Smooth Infinitesimal AnalysisJournal of Philosophical Logic 35 (6): 621651. 2006.A remarkable development in twentiethcentury mathematics is smooth infinitesimal analysis ('SIA'), introducing nilsquare and nilpotent infinitesimals, recovering the bulk of scientifically applicable classical analysis ('CA') without resort to the method of limits. Formally, however, unlike Robinsonian 'nonstandard analysis', SIA conflicts with CA, deriving, e.g., 'not every quantity is either = 0 or not = 0.' Internally, consistency is maintained by using intuitionistic logic (without the law …Read more

101995–1996 annual meeting of the association for symbolic logicBulletin of Symbolic Logic 2 (4): 448472. 1996.

How to Godel a FregeRussellIn A. D. Irvine (ed.), Bertrand Russell: Critical Assessments, Routledge. pp. 154. 1999.

60With the rise of multiple geometries in the nineteenth century, and in the last century the rise of abstract algebra, of the axiomatic method, the settheoretic foundations of mathematics, and the inﬂuential work of the Bourbaki, certain views called “structuralist” have become commonplace. Mathematics is seen as the investigation, by more or less rigorous deductive means, of “abstract structures”, systems of objects fulﬁlling certain structural relations among themselves and in relation to othe…Read more

43EPR, bell, and collapse: A route around "stochastic" hidden variablesPhilosophy of Science 54 (4): 558576. 1987.Two EPR arguments are reviewed, for their own sake, and for the purpose of clarifying the status of "stochastic" hidden variables. The first is a streamlined version of the EPR argument for the incompleteness of quantum mechanics. The role of an antiinstrumentalist ("realist") interpretation of certain probability statements is emphasized. The second traces out one horn of a central foundational dilemma, the collapse dilemma; complex modal reasoning, similar to the original EPR, is used to deri…Read more

41Realist principlesPhilosophy of Science 50 (2): 227249. 1983.We list, with discussions, various principles of scientific realism, in order to exhibit their diversity and to emphasize certain serious problems of formulation. Ontological and epistemological principles are distinguished. Within the former category, some framed in semantic terms (truth, reference) serve their purpose visavis instrumentalism (Part 1). They fail, however, to distinguish the realist from a wide variety of (constructional) empiricists. Part 2 seeks purely ontological formulatio…Read more

28Quantum logic and the projection postulatePhilosophy of Science 48 (3): 469486. 1981.This paper explores the status of the von NeumannLuders state transition rule (the "projection postulate") within "reallogic" quantum logic. The entire discussion proceeds from a reading of the Luders rule according to which, although idealized in applying only to "minimally disturbing" measurements, it nevertheless makes empirical claims and is not a purely mathematical theorem. An argument (due to Friedman and Putnam) is examined to the effect that QL has an explanatory advantage over Copenh…Read more

98On nominalismPhilosophy and Phenomenological Research 62 (3): 691705. 2001.Probably there is no position in Goodman’s corpus that has generated greater perplexity and criticism than Goodman’s “nominalism”. As is abundantly clear from Goodman’s writings, it is not “abstract entities” generally that he questions—indeed, he takes sensory qualia as “basic” in his Carnapinspired constructional system in Structure—but rather just those abstracta that are so crystal clear in their identity conditions, so fundamental to our thought, so prevalent and seemingly unavoidable in o…Read more

21The Many Worlds Interpretation of Set TheoryPSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1988 445455. 1988.Standard presentations of axioms for set theory as truths simpliciter about actualobjects the setsconfront a number of puzzles associated with platonism and foundationalism. In his classic, Zermelo suggested an alternative "many worlds" view. Independently, Putnam proposed something similar, explicitly incorporating modality. A modalstructural synthesis of these ideas is sketched in which obstacles to their formalization are overcome. Extendability principles are formulated and used to motiva…Read more

24Mathematics without Numbers: Towards a ModalStructural InterpretationPhilosophical Review 101 (4): 919. 1992.
Areas of Specialization
Aesthetics 
Logic and Philosophy of Logic 
Philosophy of Mathematics 
Philosophy of Physical Science 
Areas of Interest
17th/18th Century Philosophy 